Europeans explored & created colonies for many reasons:
1) In search of a passage to China: China was known as a "well-developed" & strong country in it's days as the Chinese empires (known as dynasty). Silk, gold, and other much needed materials came from there, and Europe wanted to open up China's market for themselves.
2) Open-space prisons: European prisons soon became overcrowded, which led to Europe shipping many of their prisoners westward.
3) Cash-crops: Later on, Europeans found that certain types of crops grew well in certain areas of the US (particularly, Tobacco, and later cotton, in the Southern part of the United States).
But most importantly:
4) Freedom of Religion: Many religious groups came from Europe to try to be able to practice their religion openly & freely, without persecution. However, while they may be open to their own, it did not mean they were open to "outside" religions that were not the original inhabitant's religion.
There may be other reasons, but these are the main ones.
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